# Isosceles Triangle Theorem

If two sides of a triangle are congruent , then the angles opposite to these sides are congruent.

$\angle P\cong \angle Q$

Proof:

Let $S$ be the midpoint of $\stackrel{\xaf}{PQ}$ .

Join $R$ and $S$ .

Since $S$ is the midpoint of $\stackrel{\xaf}{PQ}$ , $\stackrel{\xaf}{PS}\cong \stackrel{\xaf}{QS}$ .

By Reflexive Property ,

$\stackrel{\xaf}{RS}\cong \stackrel{\xaf}{RS}$

It is given that $\stackrel{\xaf}{PR}\cong \stackrel{\xaf}{RQ}$

Therefore, by SSS ,

$\Delta PRS\cong \Delta QRS$

Since corresponding parts of congruent triangles are congruent,

$\angle P\cong \angle Q$

The converse of the Isosceles Triangle Theorem is also true.

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

If $\angle A\cong \angle B$ , then $\stackrel{\xaf}{AC}\cong \stackrel{\xaf}{BC}$ .